In Modern Warfare, Outlaw sees an escorting Cobra get shot down, and despite the nuclear threat, they decide to try and rescue the pilot of the Cobra. It's pointless as the nuke goes off just seconds before the Sea Knight reaches minimum safe distance.
The impact of the scene is enhanced by the fact that Price reflects on the long history between him and Soap as various pictures of Soap and Price are shown in cheery moods during their Task Force 141 days (Ghost appears in one of these pictures too along with Sandman). Price then says regarding Soap that you don't really appreciate your friends and their accomplishments until they are gone and you try your best to keep the fond memories of them alive even as their faces fade from your memory.
In Modern Warfare 3, shortly after Soap dies, Baseplate aka MacMillan consoles Price upon hearing of what happened, even calling him "Son" and in his voice you can tell he really gives a damn about his protege, you get that fatherly vibe off him.
The fatherly vibe is present in every conversation. Note that he is the only person in any of the games to address Price by his first name, John. Even as late as the current game, the Master-Apprentice Chain remains intact.
There's a moment in MW3, second to last mission I believe as you search for the Russian prime minster. After a lengthy firefight, you duck into some tunnels and run down a ramp. As you round the corner, there are two Russians standing there. At first, this troper shot them out of reflex, but after noticing that the bodies had fallen close together and on their faces without firing back, he reloaded the checkpoint out of curiosity. Upon running down the same tunnel for a second time without shooting, the scene was this. Two soldiers, unarmed and limping, holding onto one another for support. It was such a small, human moment that he couldn't fire at them again. This was of course negated somewhat when one of the AI teammates then gunned them down without thinking. They were avenged though, as all the AI teammates are killed at the end of the level.
A small but significant Heartwarming moment appears in the briefing of "Dust to Dust" - when the logo of Task Force 141 is shown, the "DISAVOWED" ribbon is no longer blemishing the aforementioned seal of true badassery. Price did it. He won. He cleared their names of treason and acts of terror. Now if only Soap and the rest of Task Force 141 had lived to see it...
The fact that this is shown immediately after Price's Briefing Wall Shrine drives it home.